Polynomial identity ring explained

In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z, over the ring of integers in N variables X₁, X₂, ..., X_N such that

P(r _{1,} r _{2,} \ldots, r _{N)} = 0 for all N-tuples r₁, r₂, ..., r_N taken from R.

Strictly the X_i here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra.

If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1.

Every commutative ring is a PI-ring, satisfying the polynomial identity XY − YX = 0. Therefore, PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic p different from zero then it satisfies the polynomial identity pX = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.^[1]

Examples

• For example, if R is a commutative ring it is a PI-ring: this is true with

P(X _{1,X} _{2)} = X _{1X} _{2-X} _{2X} _{1} = 0 ~

• The ring of 2 × 2 matrices over a commutative ring satisfies the Hall identity

(xy - yx) ^{2z=z(xy-yx)} ^{2}

This identity was used by, but was found earlier by .

• A major role is played in the theory by the standard identity s_N, of length N, which generalises the example given for commutative rings (N = 2). It derives from the Leibniz formula for determinants

\det(A) = \sum _{\sigma \in S _{N}} sgn (\sigma) \prod ^{N} _{i =1} a _{i,\sigma(i)}

by replacing each product in the summand by the product of the X_i in the order given by the permutation σ. In other words each of the N ! orders is summed, and the coefficient is 1 or −1 according to the signature.

s _{N(X} _{1,\ldots,X} _{N)} = \sum _{\sigma \in S _{N}} sgn (\sigma) X _{\sigma(1)} ... m X _{\sigma(N)} =0 ~

The m × m matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies s_2m. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2m.

• Given a field k of characteristic zero, take R to be the exterior algebra over a countably infinite-dimensional vector space with basis e₁, e₂, e₃, ... Then R is generated by the elements of this basis and

e_i e_j = − e_j e_i.

This ring does not satisfy s_N for any N and therefore can not be embedded in any matrix ring. In fact s_N(e₁,e₂,...,e_N) = N ! e₁e₂...e_N ≠ 0. On the other hand it is a PI-ring since it satisfies x, y, z] := xyz − yxz − zxy + zyx = 0. It is enough to check this for monomials in the e_i's. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree [''x'', ''y''] := xy − yx = 0. If both are of odd degree then [''x'', ''y''] = xy − yx = 2xy has even degree and therefore commutes with z, i.e. x, y, z] = 0.

Properties

• Any subring or homomorphic image of a PI-ring is a PI-ring.
• A finite direct product of PI-rings is a PI-ring.
• A direct product of PI-rings, satisfying the same identity, is a PI-ring.
• It can always be assumed that the identity that the PI-ring satisfies is multilinear.
• If a ring is finitely generated by n elements as a module over its center then it satisfies every alternating multilinear polynomial of degree larger than n. In particular it satisfies s_N for N > n and therefore it is a PI-ring.
• If R and S are PI-rings then their tensor product over the integers,

R ⊗ _{ZS} , is also a PI-ring.

• If R is a PI-ring, then so is the ring of n × n matrices with coefficients in R.

PI-rings as generalizations of commutative rings

Among non-commutative rings, PI-rings satisfy the Köthe conjecture. Affine PI-algebras over a field satisfy the Kurosh conjecture, the Nullstellensatz and the catenary property for prime ideals.

If R is a PI-ring and K is a subring of its center such that R is integral over K then the going up and going down properties for prime ideals of R and K are satisfied. Also the lying over property (If p is a prime ideal of K then there is a prime ideal P of R such that

p is minimal over P\cap K ) and the incomparability property (If P and Q are prime ideals of R and P\subset Q then P\cap K\subset Q\cap K ) are satisfied.

The set of identities a PI-ring satisfies

If F := Z is the free algebra in N variables and R is a PI-ring satisfying the polynomial P in N variables, then P is in the kernel of any homomorphism

\tau

F

→ R.

An ideal I of F is called T-ideal if

f(I)\subset I for every endomorphism f of F.

Given a PI-ring, R, the set of all polynomial identities it satisfies is an ideal but even more it is a T-ideal. Conversely, if I is a T-ideal of F then F/I is a PI-ring satisfying all identities in I. It is assumed that I contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.

See also

• Posner's theorem
• Central polynomial

References

• • • Polynomial identities in ring theory, Louis Halle Rowen, Academic Press, 1980,
• Polynomial identity rings, Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004,
• Polynomial identities and asymptotic methods, A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005,
• Computational aspects of polynomial identities, Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005,

Further reading

• Book: Formanek, Edward . The polynomial identities and invariants of n×n matrices . 0714.16001 . Regional Conference Series in Mathematics . 78 . Providence, RI . . 1991 . 0-8218-0730-7 .
• Book: Kanel-Belov . Alexei . Rowen . Louis Halle . Computational aspects of polynomial identities . 1076.16018 . Research Notes in Mathematics . 9 . Wellesley, MA . A K Peters . 1-56881-163-2 . 2005 .

Notes and References

1. J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Vol 30